Theorem without proof: The image of a line reflected across the arc of a circle, is a circle. The image of a ray or segment is a circular arc.
I used algebra for the proof. Having guessed (observation) that the reflection was a circle, I set up the vector problem: There is a single point M and a corresponding constant c which satisfy, for any point X on the line, X' = M + c (cos θ, sin θ) The crux is this. Drop an altitude from A to the line, and draw the intersection P. Reflect P about α: [math]\;\;\;\; P' = A+ \frac{r²}{AP²} (P-A).[/math] On the reflection, P' is the closest point to the given line. Now consider a second point Z. very far from A. The further Z gets, the closer Z' is to A: [math]\;\;\;\;Z' =A+ \frac{r²}{AZ²} (Z-A) = A+ \frac{r²}{AZ} ({\bf \hat w})[/math] [math]{\bf \hat w}[/math] has constant length, and r is a constant. The denominator increases without bound. In the limit as Z gets infinitely far away, Z' = A. So both ends of the line project to point A. The figure is closed. A line doesn't make any weird deviations, it just keeps on keepin' on. The reflection should have constant curvature. A closed figure with constant curvature is a circle, and if this is all true, AP' should be a diameter. Test the midpoint M of AP' directly. Success. _________________ The Tangent Circle Problem: [list] [*]1. Tangent along the rim: solve for k [*]2a. Initial position: [url]http://www.geogebratube.org/material/show/id/58360[/url] [*]2b. Tangent to equal circles: [url]http://www.geogebratube.org/material/show/id/58455[/url] [*]3a. Four mutually tangent & exterior circles (Apollonius): [url]http://www.geogebratube.org/material/show/id/58189 [/url] [*]3b. Vector reduction: [url]http://www.geogebratube.org/material/show/id/58461[/url] [/list] [list] [*]Affine Transformation [url]http://www.geogebratube.org/material/show/id/58177[/url] [*][b]→Reflection: Line about a Circle[/b] [*]Reflection: Circle about a Circle [url]http://www.geogebratube.org/material/show/id/58185[/url] [*]Circle Inversion: The Metric Space [url]http://www.geogebratube.org/material/show/id/60132[/url] [/list] Iteration: [list] [*]Sequences 1: Formation [url]http://www.geogebratube.org/material/show/id/58896[/url] [*]Sequence 1: Formation [url]http://www.geogebratube.org/material/show/id/59816[/url] [*]Sequence 1: Iteration 1 [url]http://www.geogebratube.org/material/show/id/59828[/url] [*]Example of equivalent projections: [url]http://www.geogebratube.org/material/show/id/65754[/url] [*]Final Diagram: [url]http://www.geogebratube.org/material/show/id/65755[/url] [/list]